Suppose we have a sample of observations, each assigned a binary treatment D; 0, 1...
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Suppose we have a sample of observations, each assigned a binary treatment D; € 0, 1 with D; = 1 indicating a unit is treated and D = 0 indicating the unit is assigned control. Assume 0 < Pr(D; = 1) < 1. We observe an outcome Y; for each observation. Define potential outcomes Y;(1) and Y:(0), denoting the outcome observed for unit i if it were assigned treatment (Y(1)) or control (Y(0)) respectively. Assume that Y: (1), Y(0) are iid from the same distribution, which implies that: E[Yi(d)] =.= E[Y, (d)] = μ(d) and Var Y₁ (d)]= ... = Var[Yn(d)]=o² (d). In class we have talked about the Average Treatment Effect (ATE), which is defined as T = E[Y:(1) - Y:(0)]. Another very common effect of interest to researchers is the Average Treatment Effect on the Treated (ATT), which is defined as: T¹ = E[Y;(1) - Y₁(0)| D; = 1]. Part a (5pts): What is the interpretation of the ATT? Give your description of what this effect means and how it is different from the ATE. Part b (10pts): Assume that: 1 Consistency holds for all treatment levels: Y = Y(1)Di +Y(0) (1 - D₂) • Weak ignorability holds only for the control outcome, i.e.: Y:(0) Di, and it is not true that: Y;(1) ILD Show that consistency of all treatment levels, and weak ignorability of the control condition (the assumptions just made) are enough to identify the ATT, i.e., show that: r = E[YD₁ = 1] E[Y|D₁ = 0]. Part c (10pts): Write and simplify the difference between the ATE and the ATT under the same assumptions as Part b. What additional assumption is necessary for this difference to be 0, and for the ATT to be equal to the ATE? Why is this assumption enough? Suppose we have a sample of observations, each assigned a binary treatment D; € 0, 1 with D; = 1 indicating a unit is treated and D = 0 indicating the unit is assigned control. Assume 0 < Pr(D; = 1) < 1. We observe an outcome Y; for each observation. Define potential outcomes Y;(1) and Y:(0), denoting the outcome observed for unit i if it were assigned treatment (Y(1)) or control (Y(0)) respectively. Assume that Y: (1), Y(0) are iid from the same distribution, which implies that: E[Yi(d)] =.= E[Y, (d)] = μ(d) and Var Y₁ (d)]= ... = Var[Yn(d)]=o² (d). In class we have talked about the Average Treatment Effect (ATE), which is defined as T = E[Y:(1) - Y:(0)]. Another very common effect of interest to researchers is the Average Treatment Effect on the Treated (ATT), which is defined as: T¹ = E[Y;(1) - Y₁(0)| D; = 1]. Part a (5pts): What is the interpretation of the ATT? Give your description of what this effect means and how it is different from the ATE. Part b (10pts): Assume that: 1 Consistency holds for all treatment levels: Y = Y(1)Di +Y(0) (1 - D₂) • Weak ignorability holds only for the control outcome, i.e.: Y:(0) Di, and it is not true that: Y;(1) ILD Show that consistency of all treatment levels, and weak ignorability of the control condition (the assumptions just made) are enough to identify the ATT, i.e., show that: r = E[YD₁ = 1] E[Y|D₁ = 0]. Part c (10pts): Write and simplify the difference between the ATE and the ATT under the same assumptions as Part b. What additional assumption is necessary for this difference to be 0, and for the ATT to be equal to the ATE? Why is this assumption enough?
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Part a The Average Treatment Effect on the Treated ATT measures the average difference in outcome between treated and untreated units conditional on being treated It is defined as ATT EY1 Y0D 1 The AT... View the full answer
Related Book For
Applied Regression Analysis and Other Multivariable Methods
ISBN: 978-1285051086
5th edition
Authors: David G. Kleinbaum, Lawrence L. Kupper, Azhar Nizam, Eli S. Rosenberg
Posted Date:
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